public static Sym3x3 ComputeWeightedCovariance(int n, Vector3[] points, float[] weights)
{
// Compute the centroid.
float total = 0f;
Vector3 centroid = new Vector3(0f);
for (int i = 0; i < n; ++i)
{
total += weights[i];
centroid += weights[i] * points[i];
}
centroid /= total;
// Accumulate the covariance matrix.
Sym3x3 covariance = new Sym3x3(0f);
for (int i = 0; i < n; ++i)
{
Vector3 a = points[i] - centroid;
Vector3 b = weights[i] * a;
covariance[0] += a.X * b.X;
covariance[1] += a.X * b.Y;
covariance[2] += a.X * b.Z;
covariance[3] += a.Y * b.Y;
covariance[4] += a.Y * b.Z;
covariance[5] += a.Z * b.Z;
}
return(covariance);
}
protected ClusterFit(ColourSet colours, SquishOptions flags) : base(colours, flags)
{
// Set the iteration count.
this._IterationCount = flags.HasFlag(SquishOptions.ColourIterativeClusterFit) ? MaxIterations : 1;
// Initialise the best error.
this._BestError = new Vector4(float.MaxValue);
// Initialize the metric
var perceptual = flags.HasFlag(SquishOptions.ColourMetricPerceptual);
if (perceptual)
{
this._Metric = new Vector4(0.2126f, 0.7152f, 0.0722f, 0.0f);
}
else
{
this._Metric = new Vector4(1.0f);
}
// Get the covariance matrix.
var covariance = Sym3x3.ComputeWeightedCovariance(colours.Count, colours.Points, colours.Weights);
// Compute the principle component
this._Principle = Sym3x3.ComputePrincipledComponent(covariance);
}
private static Vector3 GetMultiplicity1Evector(Sym3x3 matrix, float evalue)
{
// Compute M
Sym3x3 m = new Sym3x3();
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// Compute U
Sym3x3 u = new Sym3x3();
u[0] = (m[3] * m[5]) - (m[4] * m[4]);
u[1] = (m[2] * m[4]) - (m[1] * m[5]);
u[2] = (m[1] * m[4]) - (m[2] * m[3]);
u[3] = (m[0] * m[5]) - (m[2] * m[2]);
u[4] = (m[1] * m[2]) - (m[4] * m[0]);
u[5] = (m[0] * m[3]) - (m[1] * m[1]);
// Find the largest component.
float mc = Math.Abs(u[0]);
int mi = 0;
for (int i = 1; i < 6; ++i)
{
float c = Math.Abs(u[i]);
if (c > mc)
{
mc = c;
mi = i;
}
}
// Pick the column with this component.
switch (mi)
{
case 0:
return(new Vector3(u[0], u[1], u[2]));
case 1:
case 3:
return(new Vector3(u[1], u[3], u[4]));
default:
return(new Vector3(u[2], u[4], u[5]));
}
}
private static Vector3 GetMultiplicity2Evector(Sym3x3 matrix, float evalue)
{
// Compute M
Sym3x3 m = new Sym3x3();
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// Find the largest component.
float mc = Math.Abs(m[0]);
int mi = 0;
for (int i = 1; i < 6; ++i)
{
float c = Math.Abs(m[i]);
if (c > mc)
{
mc = c;
mi = i;
}
}
// pick the first eigenvector based on this index
switch (mi)
{
case 0:
case 1:
return(new Vector3(-m[1], m[0], 0.0f));
case 2:
return(new Vector3(m[2], 0.0f, -m[0]));
case 3:
case 4:
return(new Vector3(0.0f, -m[4], m[3]));
default:
return(new Vector3(0.0f, -m[5], m[4]));
}
}
private static Vector3 GetMultiplicity2Evector(Sym3x3 matrix, float evalue)
{
// Compute M
var m = new Sym3x3();
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// Find the largest component.
var mc = Math.Abs(m[0]);
var mi = 0;
for (int i = 1; i < 6; ++i) {
var c = Math.Abs(m[i]);
if (c > mc) {
mc = c;
mi = i;
}
}
// pick the first eigenvector based on this index
switch (mi) {
case 0:
case 1:
return new Vector3(-m[1], m[0], 0.0f);
case 2:
return new Vector3(m[2], 0.0f, -m[0]);
case 3:
case 4:
return new Vector3(0.0f, -m[4], m[3]);
default:
return new Vector3(0.0f, -m[5], m[4]);
}
}
private static Vector3 GetMultiplicity1Evector(Sym3x3 matrix, float evalue)
{
// Compute M
var m = new Sym3x3();
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// Compute U
var u = new Sym3x3();
u[0] = (m[3] * m[5]) - (m[4] * m[4]);
u[1] = (m[2] * m[4]) - (m[1] * m[5]);
u[2] = (m[1] * m[4]) - (m[2] * m[3]);
u[3] = (m[0] * m[5]) - (m[2] * m[2]);
u[4] = (m[1] * m[2]) - (m[4] * m[0]);
u[5] = (m[0] * m[3]) - (m[1] * m[1]);
// Find the largest component.
var mc = Math.Abs(u[0]);
var mi = 0;
for (int i = 1; i < 6; ++i) {
var c = Math.Abs(u[i]);
if (c > mc) {
mc = c;
mi = i;
}
}
// Pick the column with this component.
switch (mi) {
case 0:
return new Vector3(u[0], u[1], u[2]);
case 1:
case 3:
return new Vector3(u[1], u[3], u[4]);
default:
return new Vector3(u[2], u[4], u[5]);
}
}
public static Sym3x3 ComputeWeightedCovariance(int n, Vector3[] points, float[] weights)
{
// Compute the centroid.
var total = 0f;
var centroid = new Vector3(0f);
for (int i = 0; i < n; ++i) {
total += weights[i];
centroid += weights[i] * points[i];
}
centroid /= total;
// Accumulate the covariance matrix.
var covariance = new Sym3x3(0f);
for (int i = 0; i < n; ++i) {
var a = points[i] - centroid;
var b = weights[i] * a;
covariance[0] += a.X * b.X;
covariance[1] += a.X * b.Y;
covariance[2] += a.X * b.Z;
covariance[3] += a.Y * b.Y;
covariance[4] += a.Y * b.Z;
covariance[5] += a.Z * b.Z;
}
return covariance;
}
public static Vector3 ComputePrincipledComponent(Sym3x3 matrix)
{
// Compute the cubic coefficients
var c0 =
(matrix[0] * matrix[3] * matrix[5])
+ (matrix[1] * matrix[2] * matrix[4] * 2f)
- (matrix[0] * matrix[4] * matrix[4])
- (matrix[3] * matrix[2] * matrix[2])
- (matrix[5] * matrix[1] * matrix[1]);
var c1 =
(matrix[0] * matrix[3])
+ (matrix[0] * matrix[5])
+ (matrix[3] * matrix[5])
- (matrix[1] * matrix[1])
- (matrix[2] * matrix[2])
- (matrix[4] * matrix[4]);
var c2 = matrix[0] + matrix[3] + matrix[5];
// Compute the quadratic coefficients
var a = c1 - ((1f / 3f) * c2 * c2);
var b = ((-2f / 27f) * c2 * c2 * c2) + ((1f / 3f) * c1 * c2) - c0;
// Compute the root count check;
var Q = (.25f * b * b) + ((1f / 27f) * a * a * a);
// Test the multiplicity.
if (float.Epsilon < Q)
return new Vector3(1f); // Only one root, which implies we have a multiple of the identity.
else if (Q < -float.Epsilon) {
// Three distinct roots
var theta = Math.Atan2(Math.Sqrt(Q), -.5f * b);
var rho = Math.Sqrt((.25f * b * b) - Q);
var rt = Math.Pow(rho, 1f / 3f);
var ct = Math.Cos(theta / 3f);
var st = Math.Sin(theta / 3f);
var l1 = ((1f / 3f) * c2) + (2f * rt * ct);
var l2 = ((1f / 3f) * c2) - (rt * (ct + (Math.Sqrt(3f) * st)));
var l3 = ((1f / 3f) * c2) - (rt * (ct - (Math.Sqrt(3f) * st)));
// Pick the larger.
if (Math.Abs(l2) > Math.Abs(l1))
l1 = l2;
if (Math.Abs(l3) > Math.Abs(l1))
l1 = l3;
// Get the eigenvector
return GetMultiplicity1Evector(matrix, (float)l1);
} else { // Q very close to 0
// Two roots
double rt;
if (b < 0.0f)
rt = -Math.Pow(-.5f * b, 1f / 3f);
else
rt = Math.Pow(.5f * b, 1f / 3f);
var l1 = ((1f / 3f) * c2) + rt;
var l2 = ((1f / 3f) * c2) - (2f * rt);
// Get the eigenvector
if (Math.Abs(l1) > Math.Abs(l2))
return GetMultiplicity2Evector(matrix, (float)l1);
else
return GetMultiplicity1Evector(matrix, (float)l2);
}
}
public static Vector3 ComputePrincipledComponent(Sym3x3 matrix)
{
// Compute the cubic coefficients
float c0 =
(matrix[0] * matrix[3] * matrix[5])
+ (matrix[1] * matrix[2] * matrix[4] * 2f)
- (matrix[0] * matrix[4] * matrix[4])
- (matrix[3] * matrix[2] * matrix[2])
- (matrix[5] * matrix[1] * matrix[1]);
float c1 =
(matrix[0] * matrix[3])
+ (matrix[0] * matrix[5])
+ (matrix[3] * matrix[5])
- (matrix[1] * matrix[1])
- (matrix[2] * matrix[2])
- (matrix[4] * matrix[4]);
float c2 = matrix[0] + matrix[3] + matrix[5];
// Compute the quadratic coefficients
float a = c1 - ((1f / 3f) * c2 * c2);
float b = ((-2f / 27f) * c2 * c2 * c2) + ((1f / 3f) * c1 * c2) - c0;
// Compute the root count check;
float Q = (.25f * b * b) + ((1f / 27f) * a * a * a);
// Test the multiplicity.
if (float.Epsilon < Q)
{
return(new Vector3(1f)); // Only one root, which implies we have a multiple of the identity.
}
else if (Q < -float.Epsilon)
{
// Three distinct roots
double theta = Math.Atan2(Math.Sqrt(Q), -.5f * b);
double rho = Math.Sqrt((.25f * b * b) - Q);
double rt = Math.Pow(rho, 1f / 3f);
double ct = Math.Cos(theta / 3f);
double st = Math.Sin(theta / 3f);
double l1 = ((1f / 3f) * c2) + (2f * rt * ct);
double l2 = ((1f / 3f) * c2) - (rt * (ct + (Math.Sqrt(3f) * st)));
double l3 = ((1f / 3f) * c2) - (rt * (ct - (Math.Sqrt(3f) * st)));
// Pick the larger.
if (Math.Abs(l2) > Math.Abs(l1))
{
l1 = l2;
}
if (Math.Abs(l3) > Math.Abs(l1))
{
l1 = l3;
}
// Get the eigenvector
return(GetMultiplicity1Evector(matrix, (float)l1));
}
else // Q very close to 0
// Two roots
{
double rt;
if (b < 0.0f)
{
rt = -Math.Pow(-.5f * b, 1f / 3f);
}
else
{
rt = Math.Pow(.5f * b, 1f / 3f);
}
double l1 = ((1f / 3f) * c2) + rt;
double l2 = ((1f / 3f) * c2) - (2f * rt);
// Get the eigenvector
if (Math.Abs(l1) > Math.Abs(l2))
{
return(GetMultiplicity2Evector(matrix, (float)l1));
}
else
{
return(GetMultiplicity1Evector(matrix, (float)l2));
}
}
}
